The reionization of the Universe is one of the most important topics of present day astrophysical research. The most plausible candidates for the reionization process are star-forming galaxies, which according to the predictions of the majority of the theoretical and semi-analytical models should dominate the HI ionizing background at z~3. We aim at measuring the Lyman continuum escape fraction, which is one of the key parameters to compute the contribution of star-forming galaxies to the UV background. We have used ultra-deep U-band imaging (U=30.2mag at 1sigma) by LBC/LBT in the CANDELS/GOODS-North field, as well as deep imaging in COSMOS and EGS fields, in order to estimate the Lyman continuum escape fraction of 69 star-forming galaxies with secure spectroscopic redshifts at 3.27<z<3.40 to faint magnitude limits (L=0.2L*, or equivalently M1500~-19). We have measured through stacks a stringent upper limit (<1.7% at 1sigma) for the relative escape fraction of HI ionizing photons from bright galaxies (L>L*), while for the faint population (L=0.2L*) the limit to the escape fraction is ~10%. We have computed the contribution of star-forming galaxies to the observed UV background at z~3 and we have found that it is not enough to keep the Universe ionized at these redshifts, unless their escape fraction increases significantly (>10%) at low luminosities (M1500>-19). We compare our results on the Lyman continuum escape fraction of high-z galaxies with recent estimates in the literature and discuss future prospects to shed light on the end of the Dark Ages. In the future, strong gravitational lensing will be fundamental to measure the Lyman continuum escape fraction down to faint magnitudes (M1500~-16) which are inaccessible with the present instrumentation on blank fields.
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